Understanding Square Roots

What are Square Roots?

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 × 3 = 9. Square roots are fundamental in mathematics and have applications in geometry, physics, engineering, and statistics.

Perfect vs. Approximate Roots

A perfect square is a number whose square root is an integer. Examples include 1, 4, 9, 16, 25, etc. Most numbers, however, are not perfect squares and their square roots are irrational numbers that can only be approximated. Our calculator identifies perfect squares and provides accurate approximations for non-perfect squares.

How to Use This Calculator

Enter any positive number in the input field
Select your desired decimal precision (default is 4 decimals)
Click "Calculate" to see the results
The calculator will show whether it's a perfect square or an approximation

Examples

Number	Type	Square Root
25	Perfect Square	5 (exact)
2	Approximate	≈1.4142
0.04	Perfect Square	0.2 (exact)

Common Perfect Squares

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

10² = 100

Tips & Notes

Square roots of negative numbers are not real numbers (they're complex numbers)
The calculator automatically handles decimal numbers and fractions
Higher precision settings may show more accurate results but require more computation
All numbers have two square roots: positive and negative (e.g., √9 = 3 and -3)

Frequently Asked Questions

What is a perfect square?

A perfect square is a number that can be expressed as the product of an integer with itself. For example, 16 is a perfect square because it equals 4 × 4.

How does the approximation work?

The calculator uses the Newton-Raphson method, an iterative algorithm that progressively refines the approximation until it reaches the desired precision.

Can I calculate square roots of negative numbers?

This calculator only handles real numbers. For negative numbers, you would need to work with complex numbers (which include the imaginary unit i, where i² = -1).

Why are there two square roots for positive numbers?

Because both a positive and negative number multiplied by itself will yield the same positive result. For example, both 5² and (-5)² equal 25.

What's the maximum number of decimal places supported?

The calculator supports up to 8 decimal places. Beyond this, JavaScript floating-point precision limitations may affect the results.

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